91Ó°ÊÓ

Partial fraction decomposition is a technique used to simplify complex rational expressions, making integration possible. By breaking a complicated fraction into simpler parts, solving becomes more manageable. In essence, we express the given fraction as a sum of simpler fractions. Consider the fraction \( \frac{3}{x(4-x)} \). Instead of solving it as-is, we rewrite it using simpler fractions like \( \frac{A}{x} + \frac{B}{4-x} \).

• You assume each term can be expressed in terms of its denominator.
• Multiply both sides by the common denominator to eliminate the fractions.
• Equate coefficients of like terms to find the constants \( A \) and \( B \).
• Substitute these values back into the expression.

In our exercise, we found \( A = \frac{3}{4} \) and \( B = \frac{3}{4} \). Thus, \( \frac{3}{x(4-x)} \) becomes \( \frac{3/4}{x} + \frac{3/4}{4-x} \). This method is crucial when the direct integration of the original fraction is complicated. Breaking it down into partial fractions simplifies the integration process significantly.

Integration techniques are the methods employed to find the integral, or antiderivative, of a function. In the context of differential equations, integration is used to solve for a function given its derivative. Here, after performing partial fraction decomposition, we aim to integrate each term individually.

• Use the integral of \( \frac{1}{x} \), which is \( \ln |x| + C \).
• Apply the linearity of integration: the integral of the sum is the sum of the integrals.
• Integrate each partial fraction term separately.

In the problem, the differential equation becomes \( \int dy = \int \left( \frac{3/4}{x} + \frac{3/4}{4-x} \right) dx \). By integrating, we get \( y = \frac{3}{4} \ln|x| - \frac{3}{4} \ln|4-x| + C \). Grouping the logarithms using properties of logs gives a neater solution: \( y = \frac{3}{4} \ln \left| \frac{x}{4-x} \right| + C \). Integration techniques like these allow us to systematically solve differential equations by finding antiderivatives.

First-order differential equations involve derivatives of the first degree, which means they include terms like \( y' \) or \( \frac{dy}{dx} \). Our original exercise is a prime example of such an equation, specifically in the form \( y' = f(x) \).

• Identify that it's a first-order due to only one derivative present.
• Separate variables to allow for straightforward integration.
• Integrate both sides to find the general solution.

In solving, we split \( y' = \frac{dy}{dx} = \frac{3}{x(4-x)} \) into \( dy = \frac{3}{x(4-x)} \, dx \). After separation, we executed partial fractions and integrated both sides, finally obtaining the general solution \( y = \frac{3}{4} \ln \left| \frac{x}{4-x} \right| + C \). First-order differential equations are fundamental because they often model simple real-world situations, and mastering these can significantly aid in understanding more complex systems.